= Banach-space proximal minimization
On a <Banach space> one can define the set of minimizers of $E(x)+\|x-q\|^2/2$, but existence and uniqueness need additional geometry or coercivity hypotheses. The optimality condition is $0\in\partial E(x)+\mathcal J(x-q)$, with the <duality mapping> $\mathcal J=\partial(\|\cdot\|^2/2)$. The Hilbert completion-of-square shift by a dual vector does not generalize unchanged. On $(\mathbb R^2,\|\cdot\|_1)$ with $E(x)=x_1+x_2$ and $q=0$, every $x_1,x_2\leq0$ with $x_1+x_2=-1$ is a minimizer; the vector $(-1,-1)$ is not one.
Back to article page
